I know of no IDE that has Tomcat built in, much less any particular brand of database.
IDE's use plugins to run a separately-installed Tomcat. The Eclipse
JEE edition has a pre-installed plugin (WTP), but I've always despised it and I'm using the Eclipse Tomcat Plugin from the Eclipse Marketplace myself.
There are also plugins to talk to databases from IDEs, although I do OK without them using the regular database console apps.
Spring Boot is an execution framework, though, not an IDE.
The secret of how to be miserable is to constantly expect things are going to happen the way that they are "supposed" to happen.
You can have faith, which carries the understanding that you may be disappointed. Then there's being a willfully-blind idiot, which virtually guarantees it.